Abstract

We consider a question on existence of a double series by the generalized Walsh system, which is universal in weighted spaces. In particular, we construct a weighted function and a double series by generalized Walsh system of the form with the for all , which is universal in concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.

1. Introduction

Let be a Banach space.

Definition 1. A series
is said to be universal in with respect to rearrangements, if for any the members of (1) can be rearranged so that the obtained series converges to by norm of .

Definition 2. The series (1) is said to be universal (in ) concerning subseries, if for any , it is possible to choose a subseries from (1), which converges to the by norm of .

Note that for one-dimensional case there are many papers that are devoted to the question on existence of various types of universal series in the sense of convergence almost everywhere and on a measure (see [1–10]).

Let be a fixed integer and . Recall the following definitions (see [11]).

The Rademacher system of order is defined inductively as follows. For let
and for let

The generalized Walsh system of order is defined by
and if , where , , , then

We denote the generalized Walsh system of order by . Not that is the classical Walsh system. The basic properties of the generalized Walsh system of order have been obtained by Chrestenson, Fine, Watari, Young, Vilenkin, and others (see [11–16]).

In [6–9], the existence of universal one-dimensional series by trigonometric and the classical Walsh system with respect to rearrangements and subseries in some weighted space . Some results for two-dimensional case for the classical Walsh system were obtained in [10] is proved. In this paper we consider the universality properties of a double series by the generalized Walsh system.

2. Preliminary Notes

Now we list some properties of , , which will be useful later.(i)Each th Rademacher function has period and
if , , .(ii)(iii) is a finite product of the Rademacher functions with values in . (iv)(v), is a complete orthonormal system in and it is a basis in for .

The rectangular and spherical partial sums of the double series
will be denoted by

If is a continuous function on , then we set

3. Main Results

Let us denote the generalized Walsh system of order by , . These are the main results of the paper.

Theorem 3. There exists a double series of the form
with the following property: for any number a weighted function satisfying
can be constructed so that the series (12) is universal in concerning subseries with respect to convergence in the sense of both spherical and rectangular partial sums.

Theorem 4. There exists a double series of the form (12) with the following property: for any number a weighted function with (13) can be constructed, so that the series (12) is universal in concerning rearrangements with respect to convergence in the sense of both spherical and rectangular partial sums.

Repeating the reasoning of the proof of [17, Lemma 2] we will receive the following lemma.

Lemma 5. For any given numbers , and a step function
where is an interval of the form , , there exist a measurable set and a polynomial of the form
which satisfy the following conditions:(1)(2)(3)(4)for every measurable subset of .

Then applying this Lemma we get the next one.

Lemma 6. For any numbers , , and for any square , there exists a measurable set and a polynomial of the form
with the following properties: (1)(2)(3)(4)for every measurable subset of .

Proof. We apply Lemma 5, setting
Then we can define a measurable set and a polynomial of the form
which satisfy the following conditions: (10) (20) (30) (40) for every measurable subset of .Set
and apply Lemma 5 again, setting
Then we can define a measurable set and a polynomial of the form
which satisfy the following conditions:(100) (200) (300) (400) for every measurable subset of .Set
where
By –, –, and (38), (39), we obtain
Thus, the statements – of Lemma 6 are satisfied. Now we will check the fulfillment of statement .Let , then for some we have and from (31) it follows that .Consequently taking relations , and (38), (39) for any measurable set we obtain
Similarly, for , we get
Lemma 6 is proved.

Lemma 7. For any numbers , and a step function
there exists a measurable set and a polynomial of the form
which satisfy the following conditions: for every measurable subset of .

Proof. Without any loss of generality, we assume that
are the constancy rectangular domain of , that is, where the function is constant.Given an integer , by applying Lemma 6 with , we find that there exists a measurable set and a polynomial of the form
with the following properties:
for every measurable subset of (see (49)).Then we can take
Set
where
From (51)–(53) and (56)–(58) we obtain
Then, let , then for some we have , consequently from (57) and (58) we have
In view of the conditions (51)–(54) and the equality on , for any measurable set we obtain
Similarly, for any we have
Lemma 7 is proved.

4. Proofs of the Theorems

Theorem 3 is proved similarly [10, Theorem 3], but for maintenance of integrity of this paper, here we will give the proof.

Proof of Theorem 3. Let
be a sequence of all step functions, values, and constancy interval endpoints which are rational numbers. Applying Lemma 7 consecutively, we can find a sequence of sets and a sequence of polynomials
which satisfy the following conditions:
for every measurable subset of .Denote
where
For an arbitrary number we set
It is obvious (see (67) and (72)) that and .We define a function in the following way:
where
From (68) and (70)–(74) we obtain the following: is a measurable function and
Consider .Hence, obviously we have (see (68) and (70))
It follows from (72)–(74) that for all and
Analogously for all and we have
By (65) and (72)–(74) for all we have
By (69) and (72)–(77) for all and we obtain
Analogously for all and we have (see (78))
Now we will show that the series (70) is universal in concerning subseries with respect to convergence by both spherical and rectangular partial sums.Let , that is,
It is easy to see that we can choose a function from the sequence (64) such that
Hence, we have
From (80) and (83) we get
Assume that numbers are chosen in such a way that the following condition is satisfied:
Now we choose a function from the sequence (64) such that
This with (86) implies
Hence and from (65) and (79)–(81) we obtain
where
Analogously we have
In quality subseries of the theorem we will take
From (87) and (88) we have
Let and be arbitrary natural numbers. Then for some natural number we have
Taking into account (89) and (93) for rectangular partial sums of (91) we get
Analogously for we have
where is the spherical partial sums of (91).From (96) and (97) we conclude that the series (70) is universal in concerning subseries with respect to convergence by both spherical and rectangular partial sums (see Definition 2). Theorem 3 is proved.

Remark 8. We can prove Theorem 4 by the same method used in the proof of Theorem 3.